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OAK RIDGE NATIONAL LABORATORY o p e r a t e d by

UNION CARBIDE CORPORATION NUCLEAR DIV IS ION f o r the

U.S. A T O M I C E N E R G Y C O M M I S S I O N

ORNL- TM- 3102

MSBR CONTROL STUDIES: ANALOG SIMULATION PROGRAM

W. H. Sides, Jr.

NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use a t the Oak Ridge Notionol Laboratory. I t is subject to revision or correction and therefore does not represent a final report.

~

This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that i t s use would not infringe privately owned rights.

.

. .

W 0 R N L-TM-3 102

Contract No. W-7405-eng-26

INSTRUMENTATION AND CONTROLS DIVISION

MSBR CONTROL STUD1 ES: ANALOG SIMULATION PROGRAM

W. H. Sides, Jr.

- report was prepared as zn account of work

sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, com- pleteness or usefulness of any information. apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. ,L -. -

M A Y 1971

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee

operated by UNION CARBIDE CORPORATION

for the U. S. ATOMIC ENERGY COMMISSION

~ T R l B U T l O N OF THIS DOCUMENT IS UNL[MIT F

iii

CONTENTS

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 . Description of the Plant and Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3 . Simulation of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 System Equations and Analog Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.1 Reactor Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.2 Primary Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 Steam Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.4 Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

,

W MSBR CONTROL STUDIES: ANALOG SIMULATION PROGRAM

W. H. Sides, Jr.

ABSTRACT

This report describes the mathematical model and analog computer program which were used in the preliminary study of the dynamics and control of the lOOO-MW(e) single-fluid Molten-Salt Breeder Reactor. The results and conclusions of the study were reported earlier in Control Srudies o f a 1000-Mw(e)MSBR, ORNL-TM-2927 (May 18,1970), by W. H. Sides, Jr.

W

1. INTRODUCTION

A preliniinary investigation was made of the dynamics and possible control schemes for the proposed 1 OOO-MW(e) single-fluid Molten-Salt Breeder Reactor (MSBR). For this purpose an analog computer simulation of the plant was devised. In this report the system, simulation model, and analog computer program are described. The specific transients investigated using this simulation, the results, and the conclusions were presented in another report.'

For the purposes of the analysis, the MSBR plant consisted of a graphite-moderated circulatingfuel (primary salt) reactor, a shell-and-tube heat exchanger for transferring the generated heat to a coolant (secondary salt), a shell-and-tube supercritical steam generator, and a possible control system. The analog simulation of the plant consisted of a lumped-parameter heat transfer model for the core, primary heat exchanger, and steam generator; a two-delayed-neutron-group model of the circulating-fuel nuclear kinetics with temperature reactivity feedbacks; and the external control system.

The simulation was carried out on the ORNL Reactor Controls Department analog computer. So that the model would have the maximum dynamic range, the system differential equations were not linearized, and as a result the requisite quantity of equipment required that the model be severely limited spatially to minimize the number of equations. In addition, the pressure in the water side of the steam generator, as well as in the rest of the plant, and the physical properties of the salts and water were taken to be time invariant. The temperature of the feedwater to the steam generators was also held constant.

2. DESCRIPTION OF THE PLANT AND CONTROL SYSTEM

The proposed lOOO-MW(e) MSBR steam-electric generating plant consisted of a 2250-MW(t) graphite-moderated molten-salt reactor, 4 shell-and-tube primary heat exchangers, and 16 shell-and-tube supercritical steam generators (Fig. 1). The reactor core contained two zones: a central zone, a cylinder -14.4 ft in diameter and -13 ft high with a primary-salt volume fraction of 0.132; and an outer zone, an annular region -1.25 ft thick and the same height as the central zone. The salt volume fraction in this region was 0.37. The primary salt, bearing 3 3 U and 32Th, flowed upward through the graphite core in a single pass and then to the tube side of one of four vertical single-pass primary heat exchangers, each -19 ft long, 5 ft in diameter, and constructed of Hastelloy N. The salt flow rate at design point was 9.48 X lo' lb/hr. The design-point temperature of the salt entering the core was 1050°F and that at the core outlet was 1300°F. The liquidus temperature of this salt was approximately 930°F.

The heat generated in the primary salt in the core was transferred from the tube side of the primary heat exchangers to a countercurrent secondary salt passing through the shell side. This salt flowed in a closed secondary loop to one of four horizontal supercritical steam generators. The four secondary loops,

W. H. Sides, Jr., Control Studies of a lOOO-Mw(e) MSBR, ORNL-TM-2927 (May 18, 1970).

1

Fig. 1. Flow Diagram of MSBR Plant. The quantities shown are totals for the entire plant.

. .

ORNL DWG. 71-3380 3

MANO

700'F b-2

Fig. 2. Simulation Model of Plant and Control System.

one for each primary heat exchanger, were independent of each other, with each loop supplying heat to four steam generators. Thus there was a total of 16 steam generators in the plant. The design-point flow rate of secondary salt in each loop was 1.78 X lo7 lb/hr. At the design point the secondary-salt cold-leg temperature was 850"F, and the hot-leg temperature was 1150°F. The liquidus temperature of this salt was -725°F.

The shell-and-tube supercritical steam generators were countercurrent single-pass U-tube exchangers -73 ft long and -18 in. in diameter and were constructed of Hastelloy N. Feedwater entered the steam generators at the design point at 700°F and a pressure of about 3750 psi. The outlet steam conditions at the design point were 1000°F and 3600 psi. Each steam generator produced steam at the design point at a rate of 6.30 X lo5 lb/hr. Reference 2 gives a complete description of an earlier. but quite similar. version of the steam generator and primary heat exchanger.

The load control system used in this study maintained the temperature of the steam delivered to the turbines at a design value of 1000°F during all steady-state conditions and within a narrow band around this value during plant transients. The rudimentary control system used in this simulation is shown in Fig. 2. It consisted of a reactor outlet temperature controller similar to that used successfully in the MSRE3 and a steam temperature controller.

Steam temperature control was accomplished by varying the secondary-salt flow rate. This method was chosen because of the relatively tight coupling which existed between steam temperature and secondary-salt flow rate. The measured steam temperature was compared with its set point of 1000"F, and any error caused the secondary-salt flow rate to change at a rate proportional to the error if the error was 2°F or less. If the error was greater than 2"F, the rate of change of the secondary-salt flow rate was limited to its rate of change for a 2°F error, which was approximately 1 l%/min.

2General Engineering Division Design Analysis Section, Desigii Study of a Heat Exchange System for One MSBR

J. R. Tallackson,MSRE Design and Operations Report, Part IIA: Nuclear and Process Instrumentation, ORNL-TM-I29 Concept, ORNL-TM-1545 (September 1967).

(February 1968).

4

To control the reactor outlet temperature, an external plant-load demand signal was used to obtain a reactor outlet temperature set point. The outlet temperature set point was a linear function of load demand, varying between 1125 and 1300°F for loads above 50% and between 1000 and 1125°F for loads below 50%. The measured value of the reactor inlet temperature was subtracted from the outlet temperature set point, and, since the primary-salt flow rate was constant, a reactor (heat) power set point was generated by multiplying this AT by a proportionality constant. The reactor power set point was a function of inlet temperature during a transient and thus a function of dynamic load. The measured value of reactor power (from neutron flux) was compared with the reactor power set point, and any error was fed to the control rod servo for appropriate reactivity adjustment. Under normal conditions, the control rod servo added or removed reactivity at a rate proportional to the reactor power error if the error was 1% or less. If the error was greater than I%, the addition or removal rate was limited to the rate for a l%error, which was about O.Ol%/sec. The maximum magnitude of reactivity that the simulation allowed was +I%.

The physical constants used in this simulation are summarized in Table 1. The various system volumes, masses, flow rates, etc., calculated from the constants are listed in Table 2.

1

J

5

Table 1. Physical Constants

Properties of Materials

P k (lb/ft3 ) [Btu hr-' (OF)-' ft-'1

CP [Btu Ib-' (OF)-' ]

Primary salt 0.324 207.8 at 11 75 OF

Secondary salt 0.360 117 at 1 OOOOF

Steam 726'F 6.08 22.7 750°F 6.59 11.4 850°F 1.67 6.78

IOOOOF 1.11 5.03

Hastelloy N 1000°F 0.115 548 1175'F 0.129

9.39 11.6

Graphite 0.42 115

Reactor Core

Central Zone Outer Zone

Diameter, f t Height, ft Salt volume fraction Fuel Graphite-to-salt heat transfer coefficient,

Temperature coefficients of reactivity, (OF)-' Btu hr-' f C 2 (OF)-'

Primary salt Graphite

Thermal-neutron lifetime, sec

Delayed neutron constants, p = 0.00264

i Pi 1 0.00102 2 0.00162

-

14.4 13 0.132 2 3 3 u

1065

-1.333 X loW5 +1.056 X lo-' 3.6 x 10-4

A; (sec-'

0.0 244 6 0.2245

16.9 13 0.37

Heat Exchangers

Primary

Exchanger Heat Steam Generator

Length, ft Triangular tube pitch, in. Tube OD, in. Wall thickness, in.

18.7 0.75 0.375 0.035

Heat transfer coefficients, Btu hr-' ft-2 (OF)-'

Tube-side fluid film 3500 Tube-wall conductance 3963 Shell-side fluid film 2130

72 0.875 0.50 0.077

Steam Outlet Feedwater Inlet 3590 6400 1224 1224 1316 1316

6

Table 2. Plant Parameters (Design Point)

Reactor Core

Heat flux, Btu/hr Primary-salt flow rate, Ib/hr Steady-state reactivity, po External loop transit time of primary salt, sec

Heat generation. MW(t) Salt volume fraction Active core volume, ft3 Primary-salt volume, ft3 Graphite volume, ft3 Primary salt mass, Ib Graphite mass, Ib Number of graphite elements Heat transfer area, ft2 Average primary-salt velocity, fps Core transit time of primary salt. ser

7.68 X lo9 [2250 MW(t)]

0 .OO 140 6.048

9.48 x 107

Zone I

1830 0.132 2117 279 1838 58,074 21 2,213 1466 30,077

-4.80 2.71

Zone I1

4 20 0.37 800 296 5 04 61,428 58,124 553 14,206

-1.04 12.5

Primary Heat Exchanger

Total for each of four exchanges, tube region only

Secondary-salt flow rate, Ib/hr Number of tubes Heat transfer area, f t2 Overall heat transfer coefficient. Btu hr-' f C 2 ( O F ) - '

Tube metal volume, ft3 Tube metal mass, Ib

Volume. ft3 Mass, Ib Velocity, fps Transit time, sec

1.78 x 107 6020 11,050 993 30 16,020

Primary Salt (Tube Side) Secondary Salt (Shell Side)

57 295 11,870 34,428 10.4 2.68 1.80 6.97

Steam flow rate. Ib/hr Number of tubes Heat transfer area, ft2 Tube metal volume, ft3 Tube metal mass, Ib

Volume, ft3 Mass, Ib Transit time, sec Average velocity, fps

Steam Generator

Total for each of 16 steam generators, tube region only

7.38 x 105 434 4102 22 12,203

Steam (Tube Side) Secondary Salt (Shell Side)

20 102 235 11,873 1.15 9.62

-62.8 7.50

7

3. SIMULATION OF THE SYSTEM

3.1 Model

. A spatially lumped parameter model used for the heat transfer system (Fig. 3) consisted of the reactor core, one primary heat exchanger, one steam generator, the nuclear kinetics, and a control system as shown in Fig. 2. All 4 primary heat exchangers were combined into 1 and all 16 steam generators into 1.

In the core the primary salt in the central zone was divided axially into four equal lumps, and the graphite was divided into two. The outer zone was divided equally into two primary-salt lumps and one graphite lump. Since the primary-salt density varied only slightly with temperature, the four central-zone lumps were of equal mass, as were the two outer-zone lumps. The two central-zone graphite lumps were of equal mass as well.

The mass flow rate of the primary salt in the two zones of the core was determined by the heat generation rate in each zone, so that the temperature rise of the primary salt in the two zones was equal. Thus, 81.4% of the flow passed through the central zone and 18.6% through the outer zone.

A two-delayed-neutron-group approximation of the circulating-fuel nuclear kinetics equations4 was used in the model. This allowed the delayed-neutron precursor concentration term Ci(t - T ~ ) (see Sect. 3.2.1) to be simulated directly with two of four available transport lag devices. The delayed-neutron fraction for 3U was 0.00264, and the prompt-neutron generation time was 0.36 msec. The coefficient of reactivity for the primary salt was -1.33 X lo-’ Sk/k per OF, which was divided equally among the six primary-salt lumps of the core model. The temperature coefficient for the graphite was +1.06 X lo-’ Sk/k per OF, which was divided equally among the three graphite lumps.

The model was designed to accommodate a variable flow rate of the primary salt as well as the secondary salt and steam. The required variations of film heat transfer coefficients with the various salt and steam flow rates were included.’ The film coefficient for secondary salt on the shell side of the primary heat exchanger and steam generator was proportional to the 0.6 power of the flow rate. The film coefficient for steam on the tube side of the steam generators was assumed to be proportional to the 0.8 power of the flow rate. The variation of the film coefficient in the reactor core and on the tube (primary salt) side of the primary heat exchangers decreased with flow, as shown in Fig. 4. The heat conductance across the tube wall in both exchangers was assumed to be constant.

The primary and secondary salts in the primary heat exchanger were divided axially into four equal lumps, with the tube wall represented by two lumps. As did the primary-salt density, the secondary-salt density varied only slightly with temperature, and thus the masses of the salt lumps were assumed to be equal and constant. A variable transport delay was included in the hot and cold legs of the secondary-salt loop to simulate the transport of secondary salt between the primary heat exchanger and the steam generator.

The secondary salt in the steam generator was axially divided into four lumps of equal mass, as in the primary heat exchanger. The steam on the tube side was likewise divided into four equal lumps spatially, but of une-qual mass. Under design conditions the supercritical steam density varied from 34 lb/ft3 at the feedwater inlet to 5 lb/ft3 at the steam outlet.6 The density of the steam in the lump nearest the feedwater entrance was taken as the average density in the quarter of the steam generator represented by that lump, or 22.7 lb/ft3. The densities of the remaining three steam lumps were determined in a similar manner. The

4J. MacPhee, “The Kinetics of Circulating Fuel Reactors,”Nucl. Sei. Eng. 4,588-97 (1958). ’Private communication from H. A. McLain, ORNL. 6Private communication from T. W. Pickel, ORNL.

+ OF CORE

t ----------- --_-- I I--- .- ----- SECSALT

Ts5 \

Tro

i-

i

i

i

i i

i

--- ZONE

TS6 996

SECSALT

Ts7 \

650

---------_

GRAPHITE

T83 I r-l 1 PRISALT f----l I SECSALT ;

I I Tpg E,

! ! c-

REACTOR CORE Tri

I t Tc Tss

PRIMARY HEATEXCHANGER STEAM

Fig. 3. Lumped-Parameter Model of MSBR Plant.

.

9

ORNL DWG. 71-3378

0.6 0.8 1.0 RATIO OF REDUCED FLOW R A T E TO DESIGN FLOW RATE

Fig. 4. Variation of Film Heat Transfer Coefficient with PrimarySalt Flow Rate in the Reactor Core and Primary Heat Exchanger.

axial temperature distribution in the steam was nonlinear also and was determined from the average design-point temperatures in each half of the steam generator. The specific heat of each lump was calculated from the enthalpy and temperature distributions. In the model, these resulting design-point values of density and specific heat were assumed to remain constant during part-load steady-state conditions and during all transients.

3.2 System Equations and Analog Computer Program

The spatially lumped parameter model used to describe the plant heat transfer system is shown in

3.2.1 Reactor Core. In the graphite the heat transfer equation was: Fig. 3.

where

Tgi = temperature of graphite lump i,

Tp = temperature of primary salt lump to which heat is being transferred,

Mgl =Mg2 = mass of graphite lump in core zone I,

Mg3 = mass of graphite lump in core zone 11,

Cpg = specific heat of graphite,

10

hfg = graphte-to-primary-salt heat transfer coefficient,

A,, = A g 2 = heat transfer area of lump in zone I ,

Ag3 = heat transfer area of lump in zone 11,

kgj = fraction of fission heat generated in graphite lump i,

Pr = reactor heat generation rate.

In the primary salt:

MpiCpf%=fFl CpjCTpi-1 -Tp i>+hfgApi (Tg-Tp i )+kp iPr >

where

Tpi = temperature of primary-salt lump i,

M p l = M p 2 =Mp3 = M p 4 = mass of primary-salt lump in core zone I ,

Mp5 =Mp6 = mass of primary-salt lump in core zone 11,

Cpf = specific heat of primary salt,

F1 = total primary-salt mass flow rate,

f = fraction of primary-salt mass flow rate in zone I ,

A, , = A p 2 = A p 3 = A p 4 = heat transfer area of lump in zone I ,

A,, =A, , = heat transfer area of lump in zone 11,

k . = fraction of fission heat generated in primary-salt lump i. P’

The reactor outlet temperature Tro was given by:

The heat transfer coefficient hfg varied with the primary-salt mass flow rate, as follows:

where hfgo is the design-point value of the graphite-to-salt heat transfer coefficient. A diode function generator was used to simulate the function hfg(F,), which is shown in Fig. 4 labeled “core.”

The fraction of the total primary-salt mass flow rate that passed through zone I (and the fraction of the total heat generated in zone I> was given in Sect. 3.1 as 0.814. The fraction of heat generated in the primary salt as compared with the graphite in both zones was 91.9% in the salt and 8.1% in the graphite. Thus in zone 11,

kg3 =0.186 X 0.081 =0.0151 , (4)

and

kp5 = kp6 = 0.186 x 0.919 x 0.5 = 0.0857 .

11

W In zone I,

kgl = kg2 = 0.814 X 0.081 X 0.5 = 0.0329 . (6)

The axial distribution of the reactor heat generation rate in the primary salt in core zone I was assumed to be a chopped cosine distribution with an extrapolation distance of 8 ft. Then, in the primary salt in zone I,

kpl = k p 4 =0.814X 0.919(J 0.3881~ ~ i n x d x / J . ~ ' ~ ~ 0 . 2 1 6 1 ~ 0 . 2 1 6 1 ~

= 0.1749 ,

kp2 =kp3 =0.814X 0.919($0'5n 0 . 3 8 8 1 ~ sinxdx/J0.'IT 0 . 2 1 6 1 ~

= 0.1992

The constant coefficients in the equations were calculated using Tables 1 and 2. Thus in Eq. ( l ) ,

hfg0 A d - Mgl cpg

1065 BtuX 15,039 ft2 X lb-"F X hr hr-ft2-"F X 1.06 X lo5 lb X 0.42 Btu X 3600 sec

-

= 0.0998 sec-' ,

where hfgo = design-point primary-salt-to-graphite heat transfer coefficient, and

w

kgl Pro = 0.0329 X "F X 7.68 X IO9 Btu X hr

Mgl Cpg 0.4452 X IO5 Btu X hr X 3600 sec

(7)

(9)

= 1 S77"Flsec ,

where Pro = design-point power level [2250 MW(t)] . The coefficients in the remaining graphite equations (i = 2, 3) were similarly calculated.

For the primary-salt heat transfer equation ( 2 ) the constant coefficients are

fF,,= 0.814X 9.48X lO'1bX hr

M p l 0.25 X 58,074 lb X hr X 3600 sec

= 1.476 sec-' ,

where F , = design point value of primary-salt mass flow rate,

hfzo AD 1 8.008 X IO6 Btu X lb-"F X hr .," . - - M p l Cpf 14,519 lb X hr-"F X 0.324 Btu X 3600 sec

= 0.4729 sec-' ,

12

and

k p l Pro = 0.1749 X OF X 7.68 X lo9 Btu X hr 4704 Btu X hr X 3600 sec MP1 CPf

= 79.32"FIsec.

The coefficients for the remaining primary-salt equations (i = 2-6) were similarly calculated. The unscaled heat transfer equations for the core were thus

(1 4) 5' = 0.0998 hfg(Fl ) (Tp3 - T g l ) f 1.577- Pr , dt Pr 0

(15) 82 dT = 0.0998 hfg(F1) (Tpl - Tg2) t 1 S77- Pr , dt Pr 0

(16) 5 = 0.1722 hfg(F1) (Tps -- Tg3) + 1.320- Pr , dt Pr 0

(1 7) F Pr dT * = 1.476' (Tri Fl 0 Pr 0 dt

T p l ) + 0.4729 hfg(F,) (Tg2 - T p l ) + 79.32 - ,

(18) F pr d T 2 = 1.476' (Tpl - T p 2 ) t 0.4729 hfg(F1) (Tg2 - T p l ) t 90.24- , dt Fl 0 Pr 0

(19) 3 dT = 1 . 4 7 6 1 F (Tp2 - Tp3) t 0.4729 hfg(F,) (Tgl - T p 3 ) f 90.24- Pr , Fl 0 Pr 0 dt

(20) F Pr dT 9 = 1.476 dt Fl 0 Pr 0

(Tp3 - T p 4 ) t 0.4729 hfg(F,) (Tgl - T p 3 ) t 79.32- ,

Tro = 0.814Tp4 + o.186Tp6 . (23)

A two-delayed-neutron-group approximation of the circulating-fuel point-kinetics equations was used for the nuclear behavior of the core. The equations were:

13

. where

P, = reactor heat generation rate,

Ci = modified delayed-neutron precursor concentration,

Pi = delayed-neutron fraction,

hi = delayed-neutron precursor decay constant,

1 = neutron generation time,

T~ = transit time of primary salt through core,

T~ = transit time of primary salt through external loop,

p = reactivity,

po = steady-state design point reactivity (associated with flowing fuel),

aPi = temperature coefficient of reactivity of salt lump pi,

agi = temperature coefficient of reactivity of graphite lump gi,

ATpi = variation from design-point temperature of salt lump pi,

ATgi = variation from design-point temperature of graphite lump gi,

pc = control rod (or other externally introduced) reactivity.

Variation of the primary-salt flow rate through the core varies the value of the transit times r, and T~

inversely proportional to flow rate. Provision was made in the simulation for these variations to enable study of transients in the primary-salt flow rate. The variation of the transit times with flow rate was given

bY

and

TL = TL0 (2)' >

where 7c0 = transit time of primary salt through the core at design-point flow rate,

T~~ = transit time of primary salt through the external loop at design-point flow rate,

F , IF1 = relative primary-salt flow rate.

The value of T~~ was as given in Table 2 , and the transit time of the primary salt through the active core was taken to be 3.57 sec (ref. 5 ) .

14

The negative temperature coefficient of reactivity for the primary salt was divided equally among the six primary-salt lumps in the core model. Similarly, the positive coefficient of the graphite was divided equally among the three graphite lumps.

U were as given in Table 1. The steady-state design-point reactivity po was given by4

The nuclear constants for the two-delayed-neutron-group approximation for

J

which yields

po = 0.001403 . (30)

After calculation of the various coefficients, the unscaled kinetics equations for the relative reactor power (neutron density) and relative delayed-neutron precursor concentrations are

d(pr‘pro) = 2 7 7 8 ~ - Pr - 7.333 - Pr + 1.1 00 2 C t 2.336 - c2 , dt Pr o Pr 0 Cl 0 C* 0

tO.2801 exp - - [;o ( z3]

Pr c2 Fl c2

dt Pr 0 c2 0 FlO c20 0.4326 - - 0.2245 - - 0.2801 - - 4 C 2 / C 2 0 ) =

c2 k--) t 0.2801 [- ex?? (--)] 1 c20 1 ’ (33)

6 3

i= 1 i= 1 p = 0.001403 - r] 0.2222 ATpi + 0.3520 ATgi + pc . (34)

The functions in brackets containing the exponentials in Eqs. (32) and (33) were generated by means of diode function generators.

Equations (14) through (23) and (31) through (34) formed the completelset for the thermal and neutronic behavior of the reactor core in this model.

The system variables in these equations and in those for the primary heat exchanger, steam generator, and control system given in the three following sections were scaled for the analog computer as follows.

15

The allowable computer voltage range from 0 to 100 V corresponded to a temperature range from 0 to 2000°F for all temperatures in the simulation. Therefore the computer variable representing temperature, V,, corresponded to system temperature, T, as

T T/ =- 2 0 ' (35)

For all flow rates the computer voltage range of 0 to 100 V corresponded to a relative flow-rate range of 0 to 1.25. Thus, the design-point flow rate (1 .O) corresponded to a computer voltage VF of 80 V, or

F V F = 8 @ - .

FO

The relative power level, PIP, , was similarly scaled. The computer variable representing power was

The computer variables, V, , representing the relative heat transfer coefficients, h ( n , were generated using diode function generators with the relative flow rate VF as the input. The computer variables were directly proportionzl to the relative heat transfer coefficients such that the design-point value of h(F) corresponded to a computer voltage Vh of 80 V.

The scaling for the relative delayed-neutron precursor concentration, C;:/Cio, in the nuclear kinetics equations was similar to that for flow rate and power level. The computer variable Vci was

Reactivity was scaled such that the range of the computer variable V p of 0 to 100 V corresponded to reactivity of 0 to 1 .O%, or

The resulting analog computer program for the heat transfer and nuclear kinetics of the core is shown in

3.2.2 Primary Heat Exchanger. The heat transfer model of the primary heat exchanger is shown in

Fig. 5 .

Fig. 3. For the primary-salt stream:

For the tube walls:

f

FI

I

I FI ( 8 0 . 0 0 v . D E S I G N POINT1

Fig. 5. Analog Computer Program for the Heat Transfer and Nuclear Kinetics Equations for the Core.

17

W

.

W

For the secondary-salt stream:

where

Tsi = temperature of secondary-salt lump i ,

Tti = temperature of tube wall lump i ,

Mp7 = M p s =Mp9 =Mri = mass of primary-salt lump,

Mtl =Mt2 = mass of tube wall lump,

Msl = Ms2 = Ms3 = Ms4 = mass of secondary-salt lump,

Cpt = specific heat of tube wall (Hastelloy N),

Cps = specific heat of secondary salt,

F2 = secondary-salt mass flow rate,

hf = primary-salt-to-tube-wall heat transfer coefficient,

h,, = tube-wall-to-secondary-salt heat transfer coefficient,

A , = heat transfer area of primary heat exchanger lump.

Heat transfer coefficients in the above equations were calculated as follows. The term hf in Eqs. (40) and (41) included the film resistance inside the tube and one-half of the tube wall resistance. The other half of the tube wall resistance and the outside-film resistance were included in the term h,, in Eqs. (41) and (42). When the flow rate of the primary salt was changed, the film coefficient varied with flow, as shown in Fig. 4, while the tube wall conductance was maintained constant. Similarly, the film-coefficient part of the term h,, varied as the 0.6 power of the secondary-salt flow rate, and the tube-wall-conductance part was constant. The calculation was performed on the analog computer by means of a diode function generator as

and

where hfo is the design-point primary-salt-to-tube-wall heat transfer coefficient and hspo is the design-point tube-wall-to-secondary-salt heat transfer coefficient. For example, from Table 1 :

-1

3500 3963

= 2428 Btu hr-' ft-2 ( O F ) - ' .

A diode function generator performed the following calculation:

(45)

(46)

18

where G(F, ) is the curve for the primary heat exchanger given in Fig. 4.

(40). The coefficients for Eqs. (40) through (42) were calculated using Tables 1 and 2. For example, in Eq.

(47) FIO =

Mp7

9.48 X lo7 lb X hr 0.25 X 47,480 lb X hr X 3600 sec

= 2.218 sec-' .

and

hf0Ap =

MP7 Cpf

2428 Btu X 11,050 ft2 X lb-"F X hr hr-ft2 -OF X 11,870 lb X 0.324 Btu X 3600 sec

= 1.938 sec-' .

The unscaled heat transfer equations for the primary heat exchanger were

dT F = 2 . 2 1 8 L (Tro - T p 7 ) t 1.938 hJF,) (Ttl - T p 7 ) .

d t Fl 0

dt Fl 0

dT F 2 = 2.218 1 (Tp7 - Tp8) + 1.938 hf ( F , ) (Tt l - T p 7 ) ,

Fl dt Fl 0

-- dTri - 2.218 - (Tp9 Tri)+ 1.938 h f ( F , ) (Tt2 - T p 9 ) ,

3 = 3.608 hf ( F , ) (Tp7 - T t l ) + 2.495 h,, (F2) (Ts3 - T t l ) dt

dTtz = 3.608 hf ( F , ) (Tp9 ~ Tt2 ) + 2.495 h,, (F,) (Tsl - Tt2) , dt

dT F2 dt F2 0

= 0.5737- (T, - T s l ) t 0.4159 hsp(F2) (Tt2 - T S l ) ,

dT F2 d t F2 0 2 = 0.5737 - (Tsl - T s 2 ) t 0.4159 h,, (F2) (Tt2 - T s l ) ,

(49)

(53)

(54)

(55)

19

W The scaling of these equations for the analog computer is given in Sect. 3.2.1. The analog computer program for these equations is shown in Fig. 6. 3.2.3 Steam Generator. The heat transfer model of the steam generator is shown in Fig. 3. For the

secondary-salt stream:

For the tube walls:

Mti Cpt$ = h,, A , (T,, - Tti) + hwi A , (Twi - T,i)

For the steam:

M wr . C Pwz .% dt = F, Cpwi (TWiwl - Twi) + h,, A, (Tti - T w i ) ,

where

Twi = temperature of steam (water) lump i,

Mss =Ms6 = Ms7 =Mss = mass of secondary-salt lump,

Mwi = mass of steam lump i,

Cpwi = specific heat of steam lump i,

F , = steam mass flow rate,

h,, = secondary-salt-to-tube-wall transfer coefficient,

h w l = tube-wall-to-steam heat transfer coefficient for the water inlet half of the steam generator,

h,, = tube-wall-to-steam heat transfer coefficient for the steam outlet half of the steam generator,

A , = heat transfer area of the secondary-salt and steam lumps,

A , = heat transfer area of the tube lumps.

The heat transfer coefficients in these equations were calculated in the same manner as were those in the primary heat exchanger (Sect. 3.2.2); namely, half of the tube wall resistance was included with the secondary-salt-to-tube-wall coefficient and half with the tube-wall-to-steam coefficient. The tube wall resistance was maintained constant under all flow and temperature conditions. The film coefficient on the secondary-salt (shell) side varied as the 0.6 power of the secondary-salt flow rate, and that on the tube (steam) side varied as the 0.8 power of the steam flow rate. The simulation of these coefficients was performed with a diode function generator as

1

-F, (G~.L~v- DESIGN POINT;

Fig. 6. Analog Computer Program for the Heat Transfer Equations for the Primary Heat Exchanger.

‘ .

21

W

.

.

W

where the "zero" subscript refers to the design-point value. For example, hssO was calculated as follows, using the information in Table 1 :

= 855.9 Btu hr-' ft-2 ( O F ) - ' ,

A diode function generator performed the following calculations:

The other heat transfer coefficients were similarly calculated. The coefficients of Eqs. (59) and (60) were calculated using Tables 1 and 2. For example, in Eq. (59):

F2 = Mss

7.1 1 X lo5 lb X hr 47,492 lb X hr X 3600 sec

-

= 0.4159 sec-I .

hssOAs =

Ms5 Cps

855.9 Btu X 16,408 ft2 X lb-"F X hr 47,492 lb X hr-ft2 - O F X 0.36 Btu X 3600 sec

= 0.2282 sec-'

The masses and specific heats of the steam lumps in Eq. (61) were calculated as follows. The temperature and pressure distributions of the steam in the steam generator at design point were obtained from ref. 6. At the midplane in the steam generator the temperature and pressure were, respectively, 750°F and 371 1 psi. From the steam tables, the enthalpy here was 1087 Btu/lb. At design point, the enthalpy of the water was raised from its inlet value of 770.8 Btu/lb to the steam outlet value of 1421 Btu/lb, an increase of 650.2 Btu/lb. The enthalpy rise from 770.8 to 1087 Btu/lb in the water inlet half of the steam generator was equally divided between lumps w l and w2, and that from 1087 to 1421 Btu/lb equally divided between lumps w3 and w4. Thus, 48.6% of the heat was transferred in the water inlet half and 51.4% in the steam outlet half of the steam generator. The average temperature of the steam in the water inlet half was (from ref. 6) -726°F. The average temperature in the steam outlet half was about 850°F. The design-point temperatures of lumps w l , w2, w3, and w4 were, therefore, 726, 750,850, and 1000°F respectively.

The total change in temperature of the secondary salt in the steam generator at design point was 300°F. Since the specific heat of this salt was constant, the 48.6% of the heat transferred in the steam outlet half of the steam generator represented a change in salt temperature of 145.9"F, or a temperature at the midplane of the exchanger of 995.9"F, which compared well with that given by ref. 6 (997°F). This was the design-point temperature for salt lump s6. The temperatures of lumps s5 and s7 were the average in the two halves of the exchanger, or 1073 and 923°F respectively.

The overall temperature difference for heat transfer between salt and steam in the water inlet end of the exchanger (between lumps s7 and w l ) was 197°F. In the steam end the difference (between lumpss5 and w3) was 223°F. The values of the average heat transfer coefficients for the steam generator were also

22

obtained from ref. 6. The average secondary-salt (shell side) film heat transfer coefficient and average tube wall conductance over the entire exchanger were 1316 and 1224 Btu hr-' ft-2 ( O F ) - ' respectively. The average film heat transfer coefficient on the steam (tube) side over the water inlet half of the steam generator was taken to be 6400 and that for the steam outlet half was 3590 Btu hr-' ft-2 (OF)-'. The average overall heat transfer coefficients in the water inlet half and the steam outlet half were about 577 and 539 Btu hr-' ft-2 ( O F ) - ' respectively. The heat transfer area in each half was, then, 2051 f t2 . The area of each lump was

2051 X 16 2

A, =

= 16,400 ft2 .

The enthalpy and temperature of each steam lump were used to calculate the effective specific heat. For example:

1 2

Cpwl =- (1087 - 770.8)/(726 - 700) (70)

= 6.08 Btu lb-' ( O F ) - '

The value of each Cpwi is given in Table 1. The value for the density of the steam for each of the four steam lumps was obtained from ref. 6 as the average density of the steam in each quarter of the steam generator.

The steam generator was about 72 ft long. The number of O.S-in.-OD tubes was calculated from the heat transfer area to be 434 tubes in each steam generator. From this information and the tube wall thickness and tube pitch given in Table 1, the mass and volume of the salt, metal, and steam were calculated for the tube region in each steam generator. These values are given in Table 2. The coefficients of Eq. (61) were then calculated. For example. the total flow area in the 434 tubes of each steam generator was 0.284 ft. The mass of steam lump 101 was

72 4

M,,,l = 16 X 0.284 ft2 X - ft X 22.7 Ib/ft3

= 1860 lb

Similarly,

Mw2 = 934 lb,

iZf,,,3 = 555 lb,

M,v4 ~ 4 1 2 lb.

The design-point flow rate of the steam was 1.181 X lo7 Ib/hr. Then

F,, =

M ,

1.181 X lo7 Ib X hr 1860 lb X hr X 3600 sec

= 1.764 sec-'

23

W

4

The term hwl0 was calculated in a manner similar to that for hssO [see Eq. (65)]. Its value was 1771 Btu hr-' ft-' (OF)-'. Then

h W l O A 3 = 1771 Btu X 16,400 ft2 X lb-"F X hr 1860 lb X hr-ft2- "F X 6.08 Btu X 3600 sec Mwl Cpwl

= 0.7134 sec-' .

The unscaled analog computer equations for the steam generator were:

(77) 5 = 0.4159- F2 (Th - T s s ) + 0.2282 h,, (F2) (Tt3 - T,s), dt F2 0

dT = 0.4159- F2 (T,s - Ts6) t 0.2282 h,, (F2) (Tt3 - T,s) , dt F2 0

3 = 0.4159 - F2 (Ts6 - Ts7) i- 0.2282 h,, (F2) (Tt4 - Ts7) , dt F2 0

3 = 0.4159 - F2 (Ts7 - Ts8) i- 0.2282 h,, (F2) (Tt4 - Ts7) 3

dt F2 0

(79)

!.% = 0.6950 h,, (F,) (Ts7 -. Tt4) i- 1.570 h w l (F,) (T , - T t 4 ) , (82) dt

~ = 1 . 7 6 4 - ( 7 0 0 - T , , ) t 0 . 7 1 3 4 h w l F3 (F3)(Tt4 - T w l ) , (83) dt F3 0

F3 (84) dT &=3.513- (Twl - T w 2 ) + 1.312hwl (F3)(Tt4 - T w l ) ,

dt F3 0

= 5 . 9 1 2 5 (Twz - T w 3 ) t 7.161 hw3 (F,) (Tt3 - T w 3 ) , (85) dt F3 0

2 dT = 7.969 - F3 (Tw3 - T w 4 ) + 14.48 hw3 (F,) (Tt3 - T w 3 ) . dt F3 0

Transport delays were used in the hot and cold legs of the secondary-salt loop to account for the transit time of the secondary salt between the primary heat exchanger and the steam generator. Thus, from Fig. 3 ,

Fig. 7. Analog Computer Program for the Heat Transfer Equations for the Steam Generator.

25

The values used for r1 and r2 were 14.5 and 11.9 sec, respectively, at the design-point flow rate of the secondary salt and were inversely proportional to that flow rate at off-design conditions. No transport delay was included for the flow of primary salt between the reactor core and the primary heat exchanger.

The scaling of the equations for the analog computer is given in Sect. 3.2.1. The analog computer program for the steam generator is shown in Fig. 7.

3.2.4 Control System. Steam Temperature Controller. The steam temperature was controlled by varying the secondary-salt flow rate and thus the heat input to the steam generator. An error in steam temperature caused the flow rate to change at a rate proportional to the error, that is,

5 = 4 T w 4 - 1000"F), dt (89)

where F, is the secondary-salt flow rate, Tw4 is the outlet steam temperature, and cy is the controller gain. The controller gain cy used in the simulation was approximately 5.5%/min change in flow rate for each

1°F error in steam temperature for errors of 2°F or less. For errors greater than 2"F, the rate of change of flow rate was limited to ll%/min. No optimization was carried out to obtain these values, but they produced reasonably good system response.

Reactor Outlet Temperature Controller. The reactor outlet temperature set point was determined by the plant load demand (Fig. 2). The set-point equations were:

Troset = 350Pdemand + 950 (0.5 < P < 1 .O)

and

= 250Pdemand + 1000 (0 < P < 0.5) , TrOset

where Troset is the reactor outlet temperature set point andPdemand is the fractional plant load demand. This set of equations was chosen so that the resulting steady-state reactor inlet temperature would decrease linearly with load from 1050°F at full load to 1000°F at 50% load and remain constant at 1000°F for loads below 50%.

The reactor power-level set point was proportional to the difference between the outlet temperature set point and the measured reactor inlet temperature, that is,

where Prset is the reactor power level set point, Tri is the measured reactor inlet temperature, and A is the proportionality constant.

The proportionality constant A was itself proportional to the primary-salt flow rate. The assumption of constant specific heat of the primary salt is thus implied here. However, it is not necessary to assume constant specific heat in an actual operating control system of this type. Additional circuitry may be provided to compensate for this effect, as was demonstrated in the MSRE control ~ y s t e m . ~

A reactor-power-level error was obtained by subtracting the set-point value from the measured value (from neutron flux), that is,

E=Pr--Pr . set (93)

26

This power error E was the input signal to a control rod servo described by the second-order transfer function:

(94)

where G is the controller gain. o is the bandwidth, ( is the damping factor, and U(s) is the Laplace transform of the servo output, dp,/dt.

In this simulation the bandwidth was 5 Hz and the damping factor was 0.5. These values are typical of the kind and size of servo which may be used in this control-rod-drive service. The gain of the controller G was such that, for [ E / = 1% of full power, the control reactivity addition or withdrawal rate was about O.Ol%/sec; that is,

where pc is the control reactivity. For power level errors greater than 1% of full power, the reactivity addition or withdrawal rate was

limited to O.OI$%/sec. Integration of Eq. (95) yields the value for the control reactivity in the kinetics equations of Sect. 3.2.1.

The steam flow rate F , was obtained directly t r w i the power demand, Pdemand. A 5-sec first-order lag was included to obtain some siniulation of the respnse of the steam valve:

where a is a proportionality constant and r is the time constant (5 sec).

The scaling of the equations for the analog computer is given in Sect. 3.2.1. The analog computer program for the control system part of the siw:.ilaLion is shown in Fig. 8.

Y

t . (

ORNL DWG. 71-3377

STEAM TEMPERATURE CONTROLLER

F, (80.00~ = DESIGN POINT 1

I

L

-I- --

-- -- CONTROL ROD DRIVE SERVO

REACTOR OUTLET TEMPERATURE CONTROLLER

Fig. 8. Analog Computer Program for the Control System.

.

29

OR N L-TM-3102

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94-95. T. W. McIntosh, Division of Reactor Development and Technology, U. S. Atomic Energy Commission, Washington, D.C. 20545

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